1. Tiered Math Instruction
OrRTI Project
November 20, 2009

2. Do not worry about your problems with mathematics,
I assure you mine are far greater.
-Albert Einstein

3. Objectives
Look at IES recommendations for assessment and instruction in Mathematics
Understand the major findings of the National Math Advisory Panel report and it’s implications to core curriculum
Look at possible interventions to support struggling mathematicians

4. The Math Caveat
A lit search for studies on reading disabilities studies and math disability studies from found over 600 studies in the area of reading and less than 50 for mathematics (12:1)
Specific RTI mathematics studies for a recent annotated bibliography totaled 9 studies

5. Level of Scientific Evidence
IES Recommendation
Level of Scientific Evidence
RTI Component
Universal screening (Tier I)
Moderate
Assessment:
Screening
Focus instruction on whole number for grades k-5 and rational number for grades 6-8
Low
Core/Tier 2/Tier 3
Systematic instruction
Strong
Solving word problems
Visual representations
Building fluency with basic arithmetic facts
Progress monitoring
Progress Monitoring
Use of motivational strategies

6. Assessment Recommendations
Recommendation 1: Universal Screening
Recommendation 7: Progress Monitoring

7. Recommendation 1
Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk.

8. Coherent Assessment Systems
Each type of assessment has a purpose
The design of the tool should match the purpose
What are the implications for screening tools used with all students?
Think purpose not tool
How do each of these purposes fit together?
Ben Clarke, 2009

9. Features Short duration measures (1 to 5 minute(s) fluency measures)
Note many measures that are short duration also used in progress monitoring.
Longer duration measures (untimed up to 20 minutes) often examine multiple aspects of number sense
Issue of purpose is critical to examine
Most research examines predictive validity from Fall to Spring.
Controversy on pa Jordan vs.. Fuchs …..
Ben Clarke, 2009

10. Universal Screening The Math Measures: K-1: Grades 2-5: -Secondary:
Missing Number
Quantity Discrimination
Number Identification
VanDerheyden: K-CBM
Grades 2-5:
Basic Facts
Concepts and Applications
Math Focal Points
-Secondary:
Prealgebra

11. Universal screener Missing Number K & 1 assessment
One minute assessment
Individually administered

12. Universal screener Quantity Discrimination K & 1 assessment
One Minute assessment
Individually administered

13. Universal screener Number Identification K & 1 assessment
One Minute assessment
Individually administered

14. VanDerheyden: K-CBM
Ben Clarke, 2009

15. Universal screener Computation 5th grade example 1-5 grade
Grows in complexity through the grades
Two to four Minute assessment (depending on grade)
Scored on digits correct
Group administered

16. Universal screener Monitoring Basic Skills 4th grade example 2-5 grade
Grows in complexity through the grades
Four to eight minutes (depending on grade)
Scored on correct answers (some have multiple answers)
Group administered
Fuchs, Fuchs and Hamlett

17. easy-CBM: Number and Operations
Ben Clarke, 2009

18. Example: Reflecting critical math content
easy-CBM
Items created according to NCTM Focal Points for grade level
48 items for screening (16 per focal point)
Ongoing research (not reviewed in practice guide)
Ben Clarke, 2009

19. Middle School Algebra measures
Designed by Foegen and colleagues assess pre-algebra and basic algebra skills. Administered and scored similar to Math-CBM
Math CBM Computation and Concepts and Applications
Concepts and Applications showed greater valdity in 6th, 7th, and 8th grade
Ben Clarke, 2009

20. Basic Skills (in Algebra)
60 items; 5 minutes
Problems include:
Solving basic fact equations;
Applying the distributive property;
Working with integers;
Combining like terms;
Simplifying expressions;
Applying proportional reasoning
Scoring: # of problems correct
Ben Clarke, 2009

21. Basic Pre-algebra skills
Ben Clarke, 2009

22. Math Screening & Monitoring
National Center on Student Progress Monitoring (
Intervention Central’s Math Worksheet Generator (
AIMSweb (
Monitoring Basic Skills Progress
(Fuchs, Hamlet & Fuchs, 1998)
The ABC’s of CBM (Hosp, Hosp,& Howell, 2007)
DIBELS Math (2nd year Beta)
Easy CBM

23. Universal Screening TTSD Decision Rules
K: Students receiving only “o” and/or “/” in the “Progression of Mathematics Stages” on the Progress Report are screened using CBM.
1-2: Students receiving only “1” and/or “/” in “math” on the Progress Report are screened using CBM.
3-5: Students receiving only “1,” “2,” and/or “/” in “math” on the Progress Report AND scoring below the 30th percentile on the OAKS, are screened using CBM.
Students who meet the above criteria are assessed using Curriculum Based Measurements (CBM: Missing Number for K/1 and Basic Facts for 2-5). Students scoring below the 25th percentile on CBMs are placed in Second Tier Interventions.

24. Suggestions
Have a district level team select measures based on critical criteria such as reliability, validity and efficiency.
Team should have measurement expertise (e.g. school psychologist) and mathematics (e.g. math specialist)
Set up a screening to occur twice a year (Fall and Winter)
Be aware of students who fall near the cut scores
Ben Clarke, 2009

25. Suggestions
Use the same screening tool across a district to enable analyzing results across schools
Districts may use results to determine the effectiveness of district initiatives.
May also be used to determine systematic areas of weakness and provide support in that area (e.g. fractions)
Ben Clarke, 2009

26. Suggestions
Select screening measures based on the content they cover with a emphasis on critical instructional objectives for each grade level.
Lower elementary: Whole Number
Upper elementary: Rational Number
Across grades: Computational Fluency (hallmark of MLD)
Ben Clarke, 2009

27. Suggestions
In grades 4-8, use screening measures in combination with state testing data.
Use state testing data from the previous year as the first cut in a screening system.
Can then use a screening measure with a reduced pool of students or a more diagnostic measure linked to the intervention program for a second cut.
Ben Clarke, 2009

28. Roadblocks
Resistance may be encountered in allocating time and resources to the collection of screening data.
Suggested Approach: Use data collection teams to streamline the data collection and analysis process.
It takes a significant amount of time to establish district-specific benchmarks. By the time district-specific benchmarks are established, a year could pass before at-risk readers are identified and appropriate instructional interventions begin.
Ben Clarke, 2009

29. Roadblocks
Questions may arise about testing students who are “doing fine”.
Suggested Approach: Screening all students allows the school or district to evaluate the impact of instructional approaches
Screening all students creates a distribution of performance allowing the identification of at-risk students
Ben Clarke, 2009

30. Roadblocks
Screening may identify students as at-risk who do not need services and miss students who do.
Suggested Approach: Schools should frequently examine the sensitivity and specificity of screening measures to ensure a proper balance and accurate decisions about student risk status.
Ben Clarke, 2009

31. Roadblocks
Screening may identify large numbers of students who need support beyond the current resources of the school or district.
Suggested Approach: Schools and districts should
Allocate resources to the students with the most risk and at critical grade levels
and
Implement school wide interventions to all students in areas of school wide low performance (e.g. Fractions)
Confidence intervals can be found in the assessments technical manual.
Ben Clarke, 2009

32. Recommendation 7
Monitor the progress of students receiving supplemental instruction and other students who are at risk.

33. Suggestions
Monitor the progress of tier 2, tier 3 and borderline tier 1 students at least once a month using grade appropriate general outcome measures.
Same team that worked on screening can also work on progress monitoring
Need to carefully consider capacity to model growth in the context of instructional decision making
Ben Clarke, 2009

34. TTSD Progress Monitoring
CBMs are given every other week
Trained instructional assistants will complete progress monitoring
Review trend lines every 12 weeks
We need a longer intervention period because growth on math CBMs happens in small increments
Look at rates of growth published by AIMSWeb

35. Growth trajectories for responders/non responders can be based on local and class or grade performance
Or use projected rate of growth from national norms— e.g. AIMSweb 50th %tile
Grade 1, .30 digit per week growth
Grade 3, .40 digit per week growth
Grade 5, .70 digit per week growth

36. Suggestions
Use curriculum-embedded assessments in intervention materials
Frequency of measures can vary - every day to once every week.
Will provide a more accurate index of whether or not the student is obtaining instructional objectives
Combined with progress monitoring provides a proximal and distal measuue of performance
Ben Clarke, 2009

37. Roadblocks Students within classes are at very different levels.
Suggested Approach: Group students across classes to create groups with similar needs.
Ben Clarke, 2009

38. Roadblocks
Insufficient time for teachers to implement progress monitoring.
Suggested Approach: Train paraprofessionals or other school staff to administer progress monitoring measures.
Ben Clarke, 2009

39. Math

40. Instructional/Curricular Recommendations
Recommendation 2: whole numbers/rational numbers
Recommendation 3: systematic instruction
Recommendation 4: solving word problems
Recommendation 5: visual representation
Recommendation 6: fluent retrieval of facts
Recommendation 8: motivational strategies

41. Recommendation 2
Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in K-3 and on rational numbers in grades 4-8.

43. Suggestions
For tier 2 and 3 students in grades K-3, interventions should focus on the properties of whole number and operations. Some older students would also benefit from this approach.
For tier 2 and 3 students in grades 4-8, interventions should focus on in depth coverage of rational number and advanced topics in whole number (e.g. long division).

44. Core curriculum content
Whole number: understand place value, compose/decompose numbers, leaning of operations, algorithms and automaticity with facts, apply to problem solving, use/knowledge of commutative, associative, and distributive properties,
Rational number: locate +/- fractions on number line, represent/compare fractions, decimals percents, sums, differences products and quotients of fractions are fractions, understand relationship between fractions, decimals, and percents, understand fractions as rates, proportionality, and probability, computational facility
Critical aspects of geometry and measurement: similar triangles, slope of straight line/linear functions, analyze properties of two and three dimensional shapes and determine perimeter, area, volume, and surface area
Lack of number sense is a serious problem because it interferes with algorithms and facts and prevents use of strategies to verify if solutions are reasonable.
Computational fluency is critical; dependent on automatic recall and requires fluency with standard algorithms and properties.
Source: Ben Clarke & Scott Baker Pacific Institutes for Research

45. Difficulty with fractions is pervasive and impedes further progress in mathematics

46. Recommendation 3
Instruction provided in math interventions should be explicit and systematic, incorporating modeling of proficient problem-solving, verbalization of thought processes, guided practice, corrective feedback and frequent cumulative review.

47. Suggestions
Districts should appoint committees with experts in mathematics instruction and mathematicians to ensure specific criteria are covered in-depth in adopted curriculums.
Integrate computation with problem solving and pictorial representations
Stress reasoning underlying calculation methods
Build algorithmic proficiency
Contain frequent review of mathematical principles
Contain assessments to appropriately place students in the program

48. Schema-based strategy instruction (Jitendra, 2004)
Teach student to represent quantitative
relationships graphically to solve problems.
• Use Explicit Strategies:
1. Problem Identification
2. Problem Representation
3. Problem Solution
• Be systematic: Teach one type of problem at
a time until students are proficient.
• Provide models of proficient problem solving.
Kathy Jungjahann

49. Suggestions
Ensure that intervention materials are systematic and explicit and include numerous models of easy and difficult problems with accompanying teacher think-alouds.
Provide students with opportunities to solve problems in a group and communicate problem- solving strategies.
Ensure that instructional materials include cumulative review in each session.

50. Point of Discussion
“Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computations. Results are consistent for students with learning disabilities, as well as other student who perform in the lowest third of a typical class.”
National Mathematics Advisory Panel Final Report p. xxiii

51. Roadblocks
Interventionists might not be familiar with using explicit instruction and might not realize how much practice is needed for students in tier 2 and tier 3 to master the material being taught.
Suggested Approach: Have interventionists observe lessons, practice with instructional materials, and provide them with corrective feedback on implementation

52. Roadblocks
Those teaching in the intervention might not be experts or feel comfortable with the math content.
Suggested Approach: Train interventionists to explain math content (including math concepts, vocabulary, procedures, reasoning and methods) using clear, student-friendly language.

53. Roadblocks
The intervention materials might not incorporate enough modeling, think-alouds, practice or cumulative review to improve students’ math performance.
Suggested Approach: Consider having a math specialist develop an instructional template which contains the elements of instruction identified above and which can be applied to various lessons.
If possible, have a math specialist coach new interventionists on how to use materials most effectively.

54. Recommendation 4
Interventions should include instruction on solving word problems that is based on common underlying structures.

55. Suggestions
Teach students about the structure of various problem types, how to categorize problems, and how to determine appropriate solutions.
Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems.

56. Roadblocks
Math curriculum material might not classify the problems in the lessons into problem types
Suggested Approach: Use a math specialist or a state or district curriculum guide to help identify the problem types covered in the curriculum at each level and the recommended strategies for solving them.
Students must be taught to understand a set of problem types and a reliable strategy for solving each type.

57. Roadblocks
As problems get more complex, so will the problem types and the task of discriminating among them.
Suggested Approach: Explicitly and systematically teach teachers and interventionists to identify problem types and how to teach students to differentiate one problem type from another.

58. Recommendation 5
Intervention materials should include opportunities for students to work with visual representations of mathematical ideas, and interventionists should be proficient in the use of visual representations of mathematical ideas.

59. Suggestions
Use visual representations such as number lines, arrays, and strip diagrams.
If necessary consider expeditious use of concrete manipulatives before visual representations. The goal should be to move toward abstract understanding.

60. Roadblocks
Because many curricular materials do not include sufficient examples of visual representations, the interventionist may need the help of the mathematics coach or other teachers in developing the visuals.
If interventionists do not fully understand the mathematical ideas behind the (representations), they are unlikely to be able to teach it to struggling students

61. Recommendation 6
Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

62. Suggestions
Provide 10 minutes per session of instruction to build quick retrieval of basic facts. Consider the use of technology, flash cards, and other materials to support extensive practice to facilitate automatic retrieval.
For student in K-2 grade explicitly teach strategies for efficient counting to improve the retrieval of math facts.
Teach students in grades 2-8 how to use their knowledge of math properties to derive facts in their heads.

63. “Basic” math facts are important!
Basic math facts knowledge
Difficulty in automatic retrieval of basic math facts impedes more advanced math operations
Fluency in math operations
Distinguishes between students with poor math skills to those with good skills (Landerl, Bevan, & Butterworth, 2004; Passolunghi & Siegel, 2004)

64. Point of Discussion
“the general concept of automaticity. . . is that, with extended practice, specific skills can read a level of proficiency where skill execution is rapid and accurate with little or no conscious monitoring … attentional resources can be allocated to other tasks or processes, including higher-level executive or control function”
(Goldman & Pellegrino, 1987, p. 145 as quoted in Journal of Learning Disabilities, “Early Identification of Students with Math Disabilities,” July/August p 294

65. Recommendation 8
Include motivational strategies in Tier 2 and Tier 3 interventions.

66. Suggestions
Reinforce or praise students for their effort and for attending to and being engaged in the lesson.
Consider rewarding student accomplishment.
Allow students to chart their progress and to set goals for improvement.

67. Mindset
Incorporate social and intellectual support from peers and teachers
Teach students that effort has a huge impact on math achievement

68. Big Ideas from IES
• Provide explicit and systematic instruction in problem solving. • Teach common underlying structures of word problems. • Use visual representations. • Verbalize your thought process. • Model proficient problem solving, provide guided practice, corrective feedback, and frequent cumulative review.

69. Putting it all Together for Multi-tiered Instruction
National Math Panel
Process in TTSD

70. Core curriculum and instruction
National Mathematics Advisory Panel Final Report, 2008
Curricular Content moving toward algebra
Teacher Proficiency
Conceptual Understanding
Fluency and Automaticity
Problem Solving
Interdependent and
mutually reinforcing

71. Core curriculum and instruction
Curricular Content
Depth Breadth
Focus + Coherence =

72. Linear proficiency vs. Spiraling (Closure after Exposure)

73. Learning Processes
Conceptual understanding, computational fluency and problem-solving skills are each essential and mutually reinforcing.
Effort-based learning has greater impact than the notion of inherent ability
The notion of “developmentally appropriate practices” based on age or grade level has consistently been proven to be wrong. Instead, learning is contingent on prior opportunities to learn.

74. Professional Development
Teacher induction programs have positive effects on all teachers.
Professional development is important- continue to build content knowledge as well as learning strategies.
Teachers who know the math content they are teaching, including the content before and beyond, have the most impact on student achievement.

75. Practices That Work Using formative assessments
Low achievers need explicit instruction in addition to daily core instruction
Technology supports drill practice and automaticity
Gifted students should accelerate and receive enrichment

76. So What? Now What?
What information coincided with your understanding of effective math instruction, or practices in your district?
What surprised you?
What implications does the report have for this school year? Future years?

77. Tier I in TTSD 45-90 minutes core instruction
K-12 curriculum alignment
Systematic instruction and feedback
Teach content to mastery
Focus on fractions!

78. What about interventions?
Emphasis on research-based instructional strategies (not “programs”)
Increase opportunities to practice a skill correctly
Guided practice (“I do, We do, You do”)
Correction routine

79. Tier II Interventions for Math in TTSD (Within the Core)
Kindergarten
Increased teacher attention during math
Grades 1-5
10 minutes of additional guided practice per day OR
10 minutes of Computer Assisted Instruction (CAI) per day

80. Tier II & III: Research on Best Practices Baker, Gersten, and Lee, 2002
Demonstrated, significant effects for:
Progress monitoring feedback, especially when accompanied by instructional recommendations
Peer Assisted Learning
Explicit teacher led and contextualized teacher facilitated approaches
Concrete feedback to Parents

81. Interventions
Emphasis on research-based instructional strategies (not programs)
Increase opportunities to practice a skill correctly
Guided practice (“I do, We do, You do”)
Correction routine
There are few, but an increasing number of research based curricula available

82. Intervention lists IES Best Evidence
Best Evidence

83. How to start and Next steps
As you get started consider
Focus on one grade or grade bands
Long term trajectories suggest end of K critical benchmark
May have more expertise/comfort with whole number approach
Screening before progress monitoring
Strategies for collecting data
Ben Clarke, 2009

84. Resources
NMAP
Center On Instruction - Mathematics
NCTM focal points
PIR website (Best Practices/Articles)
National Center Progress Monitoring
CA Intervention Standards
Ben Clarke, 2009

85. Discussion
From where you sit in your current job, was the presentation consistent with how you think about RtI in Math?
Why? Why not?

86. Contacts Dean Richards Jon Potter Lisa Bates drichards@ttsd.k12.or.us
Jon Potter
Lisa Bates

87. Break Time